GCSE · Maths · AQA · Spec 8300

Substitute into formulae and expressions

You don't have to understand a formula to use it. It's a set of instructions with blanks: fill each blank with its number, and the answer drops out.

Feed the machine: x² − 4

-5-2.502.55-6181522x, the number you put invalue of x² − 4(0, -4)

x, the number you put in: 0. value of x² − 4: -4

Predict first: what does x² − 4 give when x = −3? Say your answer out loud, then drag the point to x = −3 and see if the curve agrees.

Watch out: Checked your prediction? Now slide across to x = 3. Same height as x = −3: both give 5. That mirror only happens because (−3)² is (−3) × (−3) = +9, exactly like 3². If you predicted −13, you squared the 3 and kept the minus, and the curve just told you no.

Maths · Substitution

Spot the hidden multiplication

Work out the value of 4a + 3b when a = 2 and b = 7.

A student's working — which line goes wrong?

Maths · Substitution

A formula you've never met

You're given this formula: s = ut + ½at². Work out s when u = 5, t = 4 and a = −2. (s is measured in metres.)

  1. s = ut + ½at²Copy the formula exactly as it's given. You don't need to know what it's for.
  2. s = (5)(4) + ½(−2)(4)²Swap every letter for its value, each in brackets. Letters side by side are multiplied, so ut becomes (5)(4).
  3. missing step
Which line is step 3?
UK note

AQA GCSE Maths: unfamiliar formulae will be given in the question. Your job is to substitute into them, not to remember them.

Maths · Substitution

Where does this answer go wrong?

Work out the value of 5 − 2y² when y = −4.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Swap each letter for its number, in brackets, and the ordinary rules of arithmetic do the rest.

What you need to know

  • How to swap each letter for its number, including negative numbers
  • Why a substituted number goes in brackets
  • The order to work in once the numbers are in
  • How to use a formula you've never seen before when it's given to you

The big picture

Substituting means feeding numbers into an expression or formula. Each letter is replaced by its value, in brackets, and the normal order of operations gives one definite answer. The method is the same for every formula, including unfamiliar scientific ones you're given.

Key points

1Substituting means replacing each letter with the number it stands for, then working out the value.
2Put every substituted number in brackets. It matters most for negatives: (−3)² = 9, but −3² = −9.
3A number written next to a letter means multiply: if x = 4, then 3x = 3 × 4 = 12.
4Once the numbers are in, work in order: brackets, then powers, then multiplication and division, then addition and subtraction.
5Subtracting a negative is the same as adding: 25 − (−6) = 25 + 6.
6A formula you have never seen works exactly the same way. Use it as it's given; you don't need to know where it comes from.

Worked example

Problem

Work out the value of a² − 3b when a = −5 and b = −2.

⚠ Watch out

Working straight from left to right. For 3 + 2x with x = 5, doing 3 + 2 first gives 5 × 5 = 25, but multiplication comes before addition: 3 + 2(5) = 3 + 10 = 13.

🧠

Memory hook

Letter out, bracket in. Then powers, then times, then plus.

✓

Check yourself

Your turn: work out 4m − n² when m = 2 and n = −3. Brackets first, then powers, then the rest. (Answer: 4(2) − (−3)² = 8 − 9 = −1.)

Flashcards

(6)
What does it mean to substitute into an expression?
Replace each letter with the number it stands for, then work out the value.
How do you write x² when x = −3?
(−3)², which is (−3) × (−3) = 9. The negative goes in brackets so the whole number is squared.
If x = 4, what is 3x?
12. A number next to a letter means multiply: 3 × 4. It never joins up to make 34.
Once the numbers are in, what order do you work in?
Brackets, then powers, then multiplication and division, then addition and subtraction.
In ½at², which letter does the ² apply to?
Only t. It means ½ × a × t², because a power belongs to the letter it's attached to.
You're given a formula you have never seen. What do you do?
Use it exactly as given: copy it, put each value in brackets in place of its letter, then work it out in the right order.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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